Reference no: EM132333661

Assignment -

Part 1 - Grocers Market Share

The retail grocery market in a small region of the UK in 2030 is dominated by 4 companies, Adleys, Ladles, Saintasmo and Tuskos.

The estimates for the percentage market share in 2030 are:

Adleys (A)	30%
Ladles (L)	20%
Saintasmo (S)	25%
Tuskos (T)	25%
Table 1. Estimated market share 2030.

For the simplicity of this coursework task the market is restricted to customers of these four supermarkets only. Also, changes in the market owing to births, deaths and 'external' migration are not considered here - so we have a fixed total market population.

The matrix A below shows the estimated probability of a customer changing their 'main' supermarket during the year 2030. (Thus, 97% of the customers of Adleys remain loyal, while 1% of Adleys customers switch to shopping in Ladle, 0.5% switch to shopping in Saintasmo, 1.5% switch to Tuskos; and so on.)

		From A	From L	From S	From T
	To A	0.9700	0.0100	0.0500	0.0300
=	To L	0.0100	0.9500	0.0400	0.0250
	To S	0.0050	0.0100	0.9000	0.0050
	To T	0.0150	0.0300	0.0100	0.9400

Figure 1. Estimated percentage Brand movement between the four supermarkets during 2030.

Section (A): (Matrix computations and Gaussian Elimination. All steps MUST be clearly shown in your solutions, and work to 5 decimal places accuracy throughout. )

(i) Set up the 2030 market share figures for the four supermarketss (shown in Table 1) as a column vector. Then use matrix multiplication with the brand movement matrix A (as shown in Figure 1) to compute the market share of each of the four supermarkets in 2031 (based on brand movements alone). Round all answers to the nearest 0.1 %.

(ii) Use Gaussian (Gauss-Jordan) elimination (without pivoting) to compute the inverse of A.

(iii) Assume that the percentages given in A also describe the brand movements during 2029, show how matrix inversion and multiplication allow you to compute the market share for each of the four supermarkets in 2029. Hence compute the market shares for each of the four supermarkets in 2029 (based on brand movements alone).

Section (B): Matrix Operations in Microsoft's Excel (You MUST use Excel to complete this section. You are required to submit your completed Excel file via the submission link on Canvas.)

Unknown to many users, Excel can do a lot of matrix operations very efficiently, either directly or through the use of built-in matrix functions. Some of the commonly used are the functions MMULT, TRANSPOSE and MINVERSE. (You may need to investigate how these functions work in Excel. Use the Excel's online Help facility to find out more.)

Your task in Section (B) is to use an Excel spreadsheet to compute the estimated market shares in the years 2028, 2029, 2031 and 2032 (based on brand movements alone). Your final spreadsheet may look like the one shown in Figure 2 (attached).

Suggestions and notes:

(i) You may assume that the movement percentages given in A also describe the brand movements during 2028, 2029, 2031 and 2032.

(ii) Start with a blank Excel spreadsheet and in a block of cells, say B4:B7, enter the population figures for the four regions. Then enter the coefficient matrix A in H4:K7, and so on.

(iii) You MUST demonstrate that your spreadsheet is constructed using appropriate Excel functions and formulas. NO credit will be given for manually entering numbers.

(iv) You should use the results from your spreadsheet to check your own calculations in Section (A).

If you have any questions please do not hesitate to contact Dr Peter Soan.

Part 2 -

1. Consider the following ordinary differential equation

dy/dx = -2x/y , y(0) = 1

(i) Find the analytical solution to this problem.

(ii) Given that y(0.7) = 0.141421 to 6 d.p., use the modified Euler method to estimate y(0.8). Take the step size h = 0.1 and work to 5 decimal place accuracy.

(iii) Now use the 4^th order Runge-Kutta method to estimate y(0.8). As before, take y(0.7) = 0.141421, h = 0.1 and work to 5 decimal place accuracy.

(iv) What does the analytical solution give for y(0.8)? Compare the numerical solutions obtained in (ii) and (iii) with this analytical solution and comment on your results. Would you expect better answers to have been obtained if a smaller value of the step size h (and thus more steps) had been used to calculate y(0.8) starting from y(0.7) ? Explain your reasoning.

2. Consider the following ordinary differential equation

dy/dx = y + xy y(1) = 2

(i) Find the analytical solution to and hence calculate the exact value of y at x = 1.1.

(ii) Use the Taylor series method, working from the ODE above, to find the first four non-zero terms of the Taylor series solution for y about x = 1. Use your series to estimate the value of y at x = 1.1.

Work to six decimal places accuracy. Compare your answer to this with that from (i).

(iii) Use Euler's method to find the approximate solution of the differential equation defined in part (i) at x = 1.1 using just one step.

(iv) Discuss the accuracy of Euler's method and the behaviour of its error as calculations develop. How would you increase the accuracy of your numerical solution obtained using Euler's method?

(v) Find the local truncation error for the Trapezoidal rule.

y_n+1 - y_n = h/2(f_n+1 + f_n)

and hence find the order of the method. What do you expect would happen to the local errors if we were to halve the step size h used? Explain your answer.

Part 3 -

1. Consider the initial value problem

x dy/dx = y, y(a) = b.

Investigate and explain why the above initial value problem

(i) Has a unique solution if a ≠ 0.

(ii) Has no solution if a = 0 and b ≠ 0.

(iii) Has infinitely many solutions if a = b = 0.

2. Consider the following equation

dy/dt = -y(m-y)

Where y is a function of t and m is the largest digit in your ID number.

(i) Use Maple and sketch a direction field for your differential equation.

(ii) Find the equilibrium solutions and determine whether they are an asymptotically stable points or not.

(iii) Solve the differential equation and then use the solution to investigate the limit of y(t) as t → ∞.

3. Define asymptotically stable and unstable equilibrium (critical) points for autonomous differential equations.

4. Consider the matrix and evaluate the exponential matrix e^tA.

5. Consider the following system

where X = (x₁, x₂)T ∈ R², m being the largest digit in your ID number.

(i) Find the general solution of the above system.

(ii) Classify the equilibrium (critical) point at the origin as to its type and determine whether it is stable, asymptotically stable or unstable.

6. Consider the following system

x'₁ = x₂

x'₂ = -x₁ + x₂

(i) Write the above equation in the form of a matrix equation

X'(t) = BX(t)

where B is a 2 x 2 matrix whose entries are real numbers.

(ii) Use Matrix method to find a general solution of the above system.

(iii) Is the equilibrium (critical) point of this system stable? Provide a brief explanation.

Attachment:- Assignment File.rar